I must, however, return to the vaulting, having overstepped my chronology by not yet noticing another most important invention. I mean the introduction of _groin-ribs_--those narrow arches erected under the lines of intersection of the vaults. The early groins had no ribs excepting the transverse ones, or _arcs-doubleaux_; the edges at which the vaults cut one another were left _bare_, and were the weakest parts of the construction; often but faintly marked, and not necessarily lying in _planes_. In more complicated vaults, such as now became necessary, this system could scarcely be continued; and the introduction of a stone rib, under every intersection, may be viewed as the crowning fact in the development of vaulting. It is impossible to lay too much stress upon its importance, for it _changed the entire geometrical system_. Up to that time the construction of groining was wholly governed by the forms of the vaulting _surfaces_; the intersections being allowed to take their chance, and to present any irregularity of figure, while the wide surfaces of vaulting were apparently carried on mere _pins’ points_ at the springing--correct enough as a mathematical figure, but ill calculated for strength. Now, however, the _intersecting lines_ assumed the government of the construction, and the form of the _surface_ was made to accommodate itself to them. They were always in planes,[6] and always true figures--usually arcs of circles; but the _panels_ of vaulting became often irregular in their configuration, and could be twisted to meet contingent requirements without offending the eye; while the ribs, all meeting in a solid springer at the foot, brought down the pressure, and deposited it firmly upon the points of support. [Illustration: Fig. 11.]